Consider a circuit consisting of a battery with an emf εand an internal resistance of r connecte

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Consider a circuit consisting of a battery with an emf εand...

Consider a circuit consisting of a battery with an emf $\varepsilon$ and an internal resistance of $r$ connected in series with a resistor $R$ and a capacitor $C$. Show that the total energy supplied by the battery while charging the battery is equal to $\varepsilon^{2} C$.

Key Concepts

Electromotive Force (EMF)

Electromotive force is the measure of the energy per unit charge provided by a power source, such as a battery. It represents the voltage across the battery's terminals when no current is flowing and serves as the initial driving force to push electrons through the circuit.

Internal Resistance

Internal resistance is an inherent property of a battery that accounts for the losses within the battery itself. It causes some of the supplied energy to be dissipated as heat, thereby affecting the amount of energy available to other circuit components and altering the voltage under load.

RC Circuit Charging

An RC circuit is one in which a resistor and a capacitor are connected in series. During the charging process, the capacitor gradually accumulates voltage in an exponential manner governed by the time constant (RC), which is crucial for understanding the transient behavior of the circuit.

Energy Storage in Capacitors

Capacitors store energy in the electric field created between their plates, and this stored energy is quantified by the expression ½CV². Despite the energy stored being ½CV² under ideal charging, the total energy supplied by the battery is higher due to losses, highlighting the practical energy conversion process in circuits.

Conservation of Energy in Electrical Circuits

The principle of conservation of energy insists that the energy supplied by the battery must be accounted for as either stored energy in the capacitor or as dissipated energy (typically as heat) in the resistive elements. This framework is essential for analyzing the energy distribution and verifying the total energy balance in the circuit.

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